|Yes you should have mentioned the failed PSU.|
Download and run a memory test program as a first step in hardware troubleshooting.
Sounds like the PSU failure may have damaged other components.
Probably best to strip down to the bare minimums and go from there. CPU/HS/fan, 1 stick of RAM, graphics, monitor, keyboard. In your case, because you are able to POST I would also keep a mouse and an optical drive.
Boot to a live version of Linux and run it off the CD. This will eliminate the hard drive and installed OS. If the same symptoms still appear then look at the motherboard or the CPU, or both as damaged. If possible you may wish to test the CPU in another computer.